Question

state the third congruence that must be given to prove that \\( \triangle abc \cong \triangle def \\) using hl congruence theorem. \\( \overline{ac} \cong \overline{df} \\) \\( \angle a \\) is a right angle

Explanation:

Step1: Recall HL Congruence Theorem

The HL (Hypotenuse - Leg) Congruence Theorem states that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the two right triangles are congruent.

Step2: Identify the given information

We are given that \(\overline{AC}\cong\overline{DF}\). For the HL theorem, we need to know that both triangles are right - angled and the hypotenuses are congruent. Since \(\angle A\) is a right angle (given), we need to know that \(\angle D\) is also a right angle. But if we consider the hypotenuses, for \(\triangle ABC\) and \(\triangle DEF\), if \(\overline{BC}\cong\overline{EF}\), along with \(\overline{AC}\cong\overline{DF}\) (given) and \(\angle A\) and \(\angle D\) being right angles (implied by the HL context as we are using the HL theorem which is for right - triangles), we can apply the HL theorem.

Answer:

\(\overline{BC}\cong\overline{EF}\)